Viscoelastic Constitutive Behavior Research Articles

Hydraulic fracturing phase-field model in porous viscoelastic media

The mechanical behavior of shale exhibits pronounced time-dependent nonlinearity. Existing phase-field models are inadequate for evaluating the fluid-driven fracture propagation process in porous viscoelastic media. In this study, a new viscoelastic phase-field model for fluid-driven fracture propagation was introduced based on a thermodynamically consistent framework using the Maxwell–Wiechert model to describe viscoelastic constitutive behavior, with the bulk and shear moduli expanded using the Prony series. Volumetric deviatoric splits were used for both elastic and viscous energies. The fluid flow in both the fracture and the matrix adhered to Darcy's law. Additionally, the phase-field driving force considers the combined influence of rock elastic, viscous, and fluid energies. The numerical calculation iteration format was constructed using finite element discretization and the Newton–Raphson (NR) iterative method. A staggered iteration algorithm was applied to address the displacement, pressure, and phase-field problems. This method not only accurately simulates mechanical cyclic loading but also models the extension of multiple hydraulic fractures in homogeneous reservoirs and hydraulic fracture propagation in fractured reservoirs.

Methodological Aspects and Mesh Convergence in Numerical Analysis of Athermal Fiber Network Material Deformation

A balance between model complexity, accuracy, and computational cost is a central concern in numerical simulations. In particular, for stochastic fiber networks, the non-affine deformation of fibers, related non-linear geometric features due to large global deformation, and size effects can significantly affect the accuracy of the computer experiment outputs and increase the computational cost. In this work, we systematically investigate methodological aspects of fiber network simulations with a focus on the output accuracy and computational cost in models with cellular (Voronoi) and fibrous (Mikado) network architecture. We study both p and h-refinement of the discretizations in finite element solution procedure, with uniform and length-based adaptive h-refinement strategies. The analysis is conducted for linear elastic and viscoelastic constitutive behavior of the fibers, as well as for networks with initially straight and crimped fibers. With relative error as the determining criterion, we provide recommendations for mesh refinement, comment on the necessity of multiple realizations, and give an overview of associated computational cost that will serve as guidance toward minimizing the computational cost while maintaining a desired level of solution accuracy.

An embedded digital element model to simulate the wrinkle defects driven by the consolidation of composites

Understanding the mechanisms of fiber wrinkling during the compaction process is of paramount importance as fiber wrinkling significantly affects the mechanical properties of composite materials. This study introduces an embedded digital element model for simulating the consolidation process of prepreg. Flexible digital element chains are employed to simulate the reinforcing effect of fibers on resin, accounting for the viscoelastic constitutive behavior of uncured resin. The simulation captures intra-layer wrinkles and in-plane waviness defects in the consolidated prepreg, thereby uncovering the mechanism behind fiber wrinkle formation during composite material prepreg consolidation. The accuracy of the model is validated through compression experiments on flat plates and L-shaped structures. The results demonstrate that the proposed embedded digital element model accurately predicts wrinkle defects. Factors such as interlayer friction of the prepreg, consolidation temperature, and mold curvature radius significantly influence the shape and size of wrinkles.

Experimental investigation of the tire wear process using camera-assisted observation assessed by numerical modeling

This paper presents a novel experimental method to study the abrasion mechanism of car tires. It is based on the detection of microscopic movements associated with material damage (cracking) on the rubber tread. This is referred to as degrading layer relaxation. It correlates with the wear rate and, interestingly, the direction of the pattern’s movement is opposite to the lateral forces during cornering. To measure and analyze the microscopic movements, a new camera-based method with feature point matching using video stabilization was developed. Besides extensive experimental investigation, the formation and propagation of microcracks are investigated using a simplified numerical model in which a phase field approach coupled with a viscoelastic constitutive behavior is implemented in a finite element framework.

The poroviscoelastodynamic solution to Mandel's problem

Mandel's problem of quasi-static poroelasticity describes axial compression of a rectangular sample from a porous, fluid-saturated, and elastic material while allowing the pore fluid to drain from the lateral boundaries. This paper reports the closed-form analytical solution to an extension of this problem that considers dynamic response of a porous, fluid-saturated, and viscoelastic specimen with similar geometry and boundary conditions to harmonic excitation. Biot's theory of poroelastodynamics, along with frequency-domain formulation of the viscoelastic constitutive behavior, are used for this purpose. Results indicate that a quasi-static approximation of the dynamic Mandel's problem could generate significantly deviant results compared to the presented dynamic solution, even at frequencies that are substantially smaller than the first natural frequency of the specimen. In this aspect, the solution delineates the existence of four timescale categories pertaining to viscous energy dissipation within the solid and fluid phases of the tested material, as well as the specimen natural frequencies and the loading excitation frequency. Relative order of the described timescales is shown to shape the various forms that the sample frequency response may take. These variations are demonstrated via three case studies involving the brain tissue, asphalt mix, and shale. The presented poroviscoelastodynamic solution verifies previous findings from experimental studies suggesting that a disentangled exhibition of the poroelastic and viscoelastic energy dissipation mechanisms may be captured by the loss angle curves of a frequency sweep test on poroviscoelastic materials.

The role of the fiber and the bond in the hygroexpansion and curl of thin freely dried paper sheets

A computationally efficient method to study the in-plane and out-of-plane dimensional instability of thin paper sheets under the influence of moisture changes is presented. The method explicitly resolves the bonded and the free segments of fibers in the sheet, capturing the effect of anisotropic hygroexpansion at the fiber level. The method is verified against a volumetric model. The importance of longitudinal fiber hygroexpansion is demonstrated in spite of the absolute value of longitudinal hygroexpansion being an order of magnitude lower than the transverse hygroexpansion component. Finally, the method is used to demonstrate the formation of macroscopic sheet curl due to a moisture gradient in structurally uniform sheets in the absence of viscoelastic or plastic constitutive behavior and through-thickness residual stress profiles.

A viscoelastic nonlinear compressible material model of lung parenchyma – Experiments and numerical identification

Characterizing material properties of lung parenchyma is essential in order to describe and predict the mechanical behavior of the lung in health and disease. Hence, we aim to identify the viscoelastic constitutive behavior of viable lung parenchyma with a particular focus on the nonlinear, compressible, and frequency-dependent material properties. To quantify the viscoelastic material behavior of rat lung parenchyma experimentally, we performed uniaxial tension tests with different frequencies, including the whole range of physiological frequencies, in combination with full-field displacement measurements (a total of 120 tests on 30 samples of 5 rats). By means of these experimental measurements, we identified the material parameters of two viscoelastic material models applicable to large three-dimensional deformations, i.e., the standard linear solid model and the model of fractional viscoelasticity. Our aim is to identify one set of material parameters that describes the whole range of physiological frequencies; therefore, we utilized a coupled inverse analysis, which equally incorporates all different tensile tests performed on one sample. The model most suitable for the description of the viscoelastic, nonlinear, and compressible material behavior of viable rat lung parenchyma is the strain energy function Ψ=356.7Pa(I1−3)+331.7Pa(I3−1.075−1)+71.05Pa(J−23I1−3)3+5.766Pa(I313−1)6 in combination with the model of fractional viscoelasticity (τ=0.06454s,α=0.5378, and β=1.856). This material model was validated to describe the complex nonlinear and compressible viscoelastic material behavior of lung parenchyma and can be utilized in finite element simulations of the whole range of physiological frequencies. Based on this model, it will be possible to quantify the stresses and strains of lung tissue during spontaneous and artificial breathing more reliable in the future.

A fractional approach to the time-temperature dependence of dynamic viscoelastic behavior

Fractional derivative and WLF equation are effective in describing the dynamic behavior and time-temperature effect of viscoelastic damping materials, respectively. These approaches have essentially evolved from the viscoelastic constitutive behavior. Based on such intrinsic relation, a fractional time-temperature superposition principle model (FTTSPM) that integrates the fractional constitutive relation and WLF equation was proposed. The parameters of this model were determined by performing tensile and DMA tests, and the master curves at 5 °C constructed by FTTSPM and WLF equation were compared. The theoretical prediction over the extended frequency span as the master curves was made by using the fractional standard linear solid model (FSLSM) to validate FTTSPM. The numerical results show that FTTSPM conforms to the time-temperature superposition principle. The parameters α and B′ in this model denote the impact of the material and environment on the shifted factor, respectively. For the storage and loss modulus, the extended frequency obtained by FTTSPM is broader than that obtained by the WLF equation. Moreover, the evaluation of the storage and loss modulus by FTTSPM is much closer to the theoretical prediction compared with that by the WLF equation. Therefore, FTTSPM is a concise and experiment-based approach with a higher precision and greater frequency-extended capacity compared with the WLF equation. However, FTTSPM inevitably faces a vertical shift when non-thermo-rheologically simple materials are considered. The physical mechanism and practical application of FTTSPM will be examined in further research.

Reduced order models for dynamic behavior of elastomer damping devices

In the context of passive damping, various mechanical systems from the space industry use elastomer components (shock absorbers, silent blocks, flexible joints,…). Among other specificities, elastomers have frequency dependent characteristics. The computational cost of the associated numerical models, using viscoelastic constitutive behavior, may become too expensive during a design process. To solve this problem, the aim of this work is to propose an efficient reduced viscoelastic model of rubber devices. The first step is to choose an accurate material model that represents the viscoelasticity. The second step is to reduce the rubber device finite element model to a super-element that keeps its frequency dependent properties. This reduced model is first built by taking into account the fact that the device's interfaces are much more rigid than the rubber core. Therefore, kinematic constraints enforce the rigid body motion of these interfaces, reducing them to twelve degrees of freedom only (three rotations and three translations per face). The super-element is then built by condensing the core of the finite element model on the interfaces via a Component Mode Synthesis (CMS) method, adapted here to viscoelastic damping. As an application, the dynamic behavior of a structure supported by four rubber devices under harmonic loads is analyzed to show the efficiency of the proposed approach.

A nonlinear model of cell interaction with an acoustic field

A theoretical and experimental nonlinear analysis of cellular response/displacement to ultrasound excitations is presented. Linear cell models can predict the resonant frequency (fR∼5MHz), but only a nonlinear analysis can reveal the amount of mechanical energy that couples into the cell and the bifurcation behavior of the cell when it is excited near resonance. The cell dynamics is described by the nonlinear viscoelastic constitutive behavior of the cytoplasm, nucleus and their respective membranes, in the presence of a fluid with an oscillating pressure field. The method of multiple scales is used to derive the amplitude of oscillation of the cytoplasm and nucleus as a function of frequency. A major finding is the existence of multiple solutions for a range of sub-resonant frequencies. At positive detuning (f>fR), the mechanical energy that couples into the cell is small, it is higher at resonance but significantly higher at sub-resonant frequencies in the multiplicity range. Experimentally it was shown when 3.5MHz is approached sub- and supra-resonance and 6.5MHz is approached sub-resonance, gene expression was statistically higher than that when stimulated directly. Thus, there exists an optimal range of frequencies for ultrasound treatment – in the region of multiplicity where deformation and thus mechanical energy coupling is maximized. The ultrasound protocol must be designed to operate at the solution associated with the higher mechanical energy – thus the start-up conditions should be in the domain of attraction of the high energy solution.

Criteria for extensional necking instability in complex fluids and soft solids. Part II: Imposed tensile stress and force protocols

We study the necking of a filament of complex fluid or soft solid subject to uniaxial tensile stretching, under conditions of constant imposed stress and force, by means of linear stability analysis and nonlinear simulations. We demonstrate necking to be a flow instability that is an unavoidable consequence of the constitutive behaviour of essentially any viscoelastic material. We derive criteria for the onset of necking that can be reported in terms of characteristic signatures in the shapes of the experimentally measured material functions, and that should therefore apply universally to all materials. To confirm their generality, we show them to hold numerically in six constitutive models. Under conditions of constant stress, we find two distinct dynamical regimes as a function of time. In the first regime the strain rate quickly attains a value prescribed by the fluid's underlying homogeneous constitutive curve, at the given stress. In this first regime, no appreciable necking arises. A second regime then ensues in which the homogeneous flow destabilises to form a neck. This necking instability can occur via two distinct modes. The first mode is relatively gentle and arises in any regime where the slope of the constitutive curve is positive. It has a rate of necking per accumulated strain unit set by the inverse of the slope of the constitutive curve. The second mode sets in when a carefully defined `elastic derivative' of the tensile force first slopes down as a function of the time. We discuss the way in which these modes of instability manifest themselves in entangled polymeric fluids, demonstrating four distinct regimes of necking behaviour as a function of stress. Under conditions of constant imposed force, typically the flow sweeps up the underlying constitutive curve of the fluid in question, with instability to necking in any regime where that curve is positively sloping.

Reduced Order Models for Dynamic Behavior of Elastomer Damping Devices

In the context of passive damping, various mechanical systems from the space industry use elastomer components (shock absorbers, silent blocks, flexible joints...). The material of these devices has frequency, temperature and amplitude dependent characteristics. The associated numerical models, using viscoelastic and hyperelastic constitutive behaviour, may become computationally too expensive during a design process. The aim of this work is to propose efficient reduced viscoelastic models of rubber devices. The first step is to choose an accurate material model that represent the viscoelasticity. The second step is to reduce the rubber device finite element model to a super-element that keeps the frequency dependence. This reduced model is first built by taking into account the fact that the device's interfaces are much more rigid than the rubber core. To make use of this difference, kinematical constraints enforce the rigid body motion of these interfaces reducing the rubber device model to twelve dofs only on the interfaces (three rotations and three translations per face). Then, the superelement is built by using a component mode synthesis method. As an application, the dynamic behavior of a structure supported by four hourglass shaped rubber devices under harmonic loads is analysed to show the efficiency of the proposed approach.

Dynamics of multilayer, multidisc viscoelastic rotor – An operator based higher order classical model

Inherent material damping plays a very significant role on dynamic behaviour of rotors. The material damping in a spinning rotor produces a tangential force along the whirl direction and its magnitude being proportional to spin speed. After certain value of spin speed, decided by the characteristic of the system, the tangential force becomes strong enough to throw the rotor centre out of the whirl orbit by inflating it progressively. This leads to destabilization of the system and corresponding speed is known as stability limit of spin speed. Stability limit of spin speed for Jeffcott rotor, by using viscous form of material damping model is straight forward and has been reported by several researchers, however the same analysis for viscoelastic material characteristics is not reported much. This analysis is very relevant for industrial requirements to replace bulky and heavy metal rotor by light but strong rotors. This is achieved either by reinforcing fibre or multi layering arrangements. Both of which are represented by viscoelastic constitutive behaviour. This paper gives mathematical derivation of equations of motion for multi-disc, multi-layered rotor–shaft-system. Both lumped mass and discretized approach (finite element) are presented here mathematically and numerical simulation results are compared. The lumped mass approach gives a concise yet acceptable accuracy of the results.

Unified viscoelasticity: Applying discrete element models to soft tissues with two characteristic times

Discrete element models have often been the primary tool in investigating and characterising the viscoelastic behaviour of soft tissues. However, studies have employed varied configurations of these models, based on the choice of the number of elements and the utilised formation, for different subject tissues. This approach has yielded a diverse array of viscoelastic models in the literature, each seemingly resulting in different descriptions of viscoelastic constitutive behaviour and/or stress-relaxation and creep functions. Moreover, most studies do not apply a single discrete element model to characterise both stress-relaxation and creep behaviours of tissues. The underlying assumption for this disparity is the implicit perception that the viscoelasticity of soft tissues cannot be described by a universal behaviour or law, resulting in the lack of a unified approach in the literature based on discrete element representations. This paper derives the constitutive equation for different viscoelastic models applicable to soft tissues with two characteristic times. It demonstrates that all possible configurations exhibit a unified and universal behaviour, captured by a single constitutive relationship between stress, strain and time as: σ+Aσ̇+Bσ¨=Pε̇+Qε¨. The ensuing stress-relaxation G(t) and creep J(t) functions are also unified and universal, derived as G(t)=c1e−A+A2−4B2Bt+(σ0−c1)e−A−A2−4B2Bt and J(t)=c2+(ε0−c2)e-PQt+σ0Pt, respectively. Application of these relationships to experimental data is illustrated for various tissues including the aortic valve, ligament and cerebral artery. The unified model presented in this paper may be applied to all tissues with two characteristic times, obviating the need for employing varied configurations of discrete element models in preliminary investigation of the viscoelastic behaviour of soft tissues.

Bayesian uncertainty analysis of finite deformation viscoelasticity

The viscoelasticity of the dielectric elastomer, VHB 4910, is experimentally characterized, modeled, and analyzed using Bayesian uncertainty analysis. Whereas these materials are known for their large-field induced deformation and broad applications in smart structures, the rate-dependent viscoelastic effects are not well understood. To address this issue, we quantify both the hyperelastic and viscoelastic constitutive behavior and use Bayesian uncertainty analysis to assess several key modeling attributes. Specifically, we compare an Ogden-based phenomenological model to a nonaffine hyperelastic model and couple hyperelasticity to both linear and nonlinear viscoelasticity. The utilization of Bayesian statistics is shown to provide insight into quantifying nonlinear viscoelasticity behavior as a function of internal state variables. The results are validated experimentally in the finite deformation regime over a range of stretch rates spanning four orders of magnitude (6.7×10-5–0.67Hz). A unique set of hyperelastic parameters are identified, independent of the stretch rate. In addition, comparisons of the linear and nonlinear viscoelastic models demonstrate a reduction in modeling error by approximately a factor of three. Finally, the viscoelastic time constant is shown to produce an inverse stretch rate power law dependence regardless of which hyperelastic model is used.

Exact solutions of isotropic viscoelastic functionally graded Kirchhoff plates

Viscoelastic equilibrium problems of Kirchhoff plates can be solved in a closed form only under special geometric assumptions, loading conditions and kinematic constraints on the boundary. A new solution procedure, based on a correspondence principle between a linearly elastic, homogeneous and orthotropic Saint-Venant beam under torsion and an isotropic linearly viscoelastic and functionally graded Kirchhoff plate with no kinematic constraints on the boundary, is proposed. The methodology is adopted to evaluate displacement, bending–twisting curvature and moment fields of an elliptic plate, with viscoelastic constitutive behavior and loading conditions described by convolution integrals, assessing thus new benchmarks for computational mechanics. The analysis is specialized to periodic fiber-reinforced composites with polymeric matrix described by a four-parameter viscoelastic model.

Thermal Expansion Coefficient Prediction of Asphalt Mixture with the Eshelby Equivalent Inclusion Theory

Asphalt mixture was considered as a two-phase composite, in which coarse aggregates are embedded into asphalt mastic matrix, namely a mix of fine aggregates and asphalt, so that a theoretical framework was proposed to correlate its effective thermal expansion coefficient with its components and microstructures based on the Eshelby equivalent inclusion theory. A four-parameter model with the experimentally determined parameters was used to characterize the viscoelastic constitutive behavior of asphalt mastic. The thermal expansion coefficient prediction of asphalt mixture was conducted and compared with the predictions by the sparse method and the self-consistent method. It was revealed that the prediction from the proposed theoretical framework is reasonable.

Modeling and simulation of relaxation in viscoelastic open cell materials and structures

Open cell materials with cubic anisotropy and structures made thereof are investigated with respect to their linear viscoelastic properties, in particular their relaxation behavior. The study is concerned with the prediction of the effective behavior which results from the isotropic bulk material properties as well as the cellular architecture. Finite Element Method simulations of three-dimensional structures are employed to predict the effective response to a wide range of loading modes in the time domain.For predicting the properties of the cellular materials and structures by the Finite Element Method different modeling strategies are employed. The first approach is a periodic unit cell method modeling an infinite medium by means of periodic boundary conditions. This way the entire effective linear viscoelastic constitutive behavior can be computed. However, it is not possible to capture effects as being attributed to traction free faces and load introduction in specimens or structures. A second approach follows to account for these effects by generating finite sample models to represent situations which occur in experimental testing. Finally, an analytical constitutive material law is developed to model linear viscoelasticity for cubic anisotropy in the time domain. It is implemented into the commercial Finite Element software ABAQUS/Standard and the material parameters are gained from the unit cell investigations. This enables the simulation of structures, parts, and components which consist or contain such cellular materials.

Thermodynamic‐based model for coupling temperature‐dependent viscoelastic, viscoplastic, and viscodamage constitutive behavior of asphalt mixtures

SUMMARYBased on the continuum damage mechanics, a general and comprehensive thermodynamic‐based framework for coupling the temperature‐dependent viscoelastic, viscoplastic, and viscodamage behaviors of bituminous materials is presented. This general framework derives systematically Schapery‐type nonlinear viscoelasticity, Perzyna‐type viscoplasticity, and a viscodamage model analogous to the Perzyna‐type viscoplasticity. The resulting constitutive equations are implemented in the well‐known finite element code Abaqus via the user material subroutine UMAT. A systematic procedure for identifying the model parameters is discussed. Finally, the model is validated by comparing the model predictions with a comprehensive set of experimental data on hot mix asphalt that include creep‐recovery, creep, uniaxial constant strain rate, and repeated creep‐recovery tests in both tension and compression over a range of temperatures, stress levels, and strain rates. Comparisons between model predictions and experimental measurements show that the presented constitutive model is capable of predicting the nonlinear behavior of asphaltic mixes under different loading conditions. Copyright © 2011 John Wiley & Sons, Ltd.

A nonlinear viscoelastic fracture analysis of concrete/FRP delamination in aggressive environments

In this paper, the durability of the bondline between concrete and FRP reinforcement was characterized at various temperature and humidity levels. The linear and nonlinear viscoelastic constitutive behavior of the epoxy bondline was characterized and used for a nonlinear viscoelastic fracture analysis of delamination. A hygrothermal nonlinear viscoelastic pseudo-stress model was developed and calibrated in order to compute a generalized J integral. Driven wedge tests were conducted for examining the fracture behavior of the interface. A finite element analysis was developed for determining the cohesive zone size and the generalized J integral at various temperature and humidity levels. The fracture energy obtained from these parameters greatly depended upon crack growth rate, temperature and humidity.